Generalized Turán numbers

Determine the generalized Turán numbers obtained by replacing the forbidden triangle and counted pentagon in the sparse pentagon theorem with arbitrary graphs, including the corresponding extremal asymptotic number of copies of the counted graph in graphs avoiding the forbidden graph.

Background

The paper proves a sparse analogue of the theorem giving the maximum number of copies of C5C_5 in a triangle-free graph. It then observes that replacing K3K_3 and C5C_5 by other graphs leads to generalized Turán-number questions. The general theory and exact values of these quantities are not known.

References

More generally, replacing~$K_3$ and~$C_5$ by other graphs in Theorem~\ref{thm:pentagon} leads to questions involving the so-called generalised Turán numbers. Results in the dense regime for this type of questions can be transferred analogously to the sparse setting using our transference principle. Much about generalised Turán numbers remains open, but see for example for a survey on what is known.

— A generalised transference principle  (2608.17982 - Allen et al., 18 Aug 2026) in Section 1, subsection “Applications”, subsection “Turán-type results”